Drummer Lee Rigby. A postmortem examination has confirmed that he died from multiple cut and stab wounds. Photograph: MoD/PA

The family of Lee Rigby, the soldier murdered in the street in Woolwich, south-east London, last week, has said he would not have wanted his death to be used to justify retaliatory attacks.

The statement, which said family members had been amazed at the "truly staggering" support they had received since Rigby was killed, was released on Friday as a coroner opened the inquest into the 25-year-old's death.

One of the suspects, Michael Adebowale, has been charged with murder and possessing a firearm, a 9.4mm Dutch calibre KNIL Model 91 revolver.

The Metropolitan police said the arrested man was 42 years old and was detained at an address in north London. The murder investigation has sent officers to search two homes in north and east London.

The Rigby family said: "We would like to emphasise that Lee would not want people to use his name as an excuse to carry out attacks against others. We would not wish any other families to go through this harrowing experience and appeal to everyone to keep calm and show their respect in a peaceful manner."

In the statement from his mother and stepfather Lyn and Ian, wife Rebecca and son Jack, among others, they said the soldier "had many friends from different walks of life – some with different religious beliefs and cultures. But this made no difference to Lee – he always treated others with the greatest of respect."

Far-right groups such as the English Defence League and the British National party have been accused of exploiting the tragedy to stir up tensions. The EDL held a rally in Woolwich, hours after Rigby was killed, and another at Downing Street on Monday. The BNP has a march planned for Saturday.

Rigby's regiment echoed the sentiments of the dead soldier's family, distancing itself from "any organisations which seek to exploit the death of one of our regimental family for a range of self-serving and unhelpful reasons. It is wrong and disgraceful that the death of one of our own should be exploited in this manner and that dishonour, by wrongful association, is brought onto the regiment." The statement, posted on Facebook, said that a number of retired soldiers were being approached and had agreed to take part in demonstrations connected to his killing.

At the opening of the inquest into Rigby's death, Deputy Chief Inspector Grant Mallon of the Metropolitan police told Southwark coroners court that the soldier was returning to the Royal Artillery barracks on 22 May after being on duty at the Tower of London when two men drove a car on to the carriageway where he was walking before attacking him with knives and a cleaver.

Mallon said one man had been charged with murder and police were waiting to interview the other. The hearing lasted approximately five minutes before it was adjourned by the deputy coroner, Lorna Tagliavini, pending further investigations.

Before concluding the hearing, Tagliavini said: "I would like to take the opportunity to offer my personal condolences to all members of Lee Rigby's family for their very sad loss."

A postmortem examination found Rigby's cause of death to be multiple cut and stab wounds. Michael Adebowale, 22, appeared in court on Thursday charged with the soldier's murder. A trial date for Adebowale has been set for 28 June.

Another suspect, Michael Adebolajo, 28, who was arrested at the scene after being shot by armed officers, is still being treated in hospital and is too ill to be questioned by detectives. Eight other people have been arrested in connection with Rigby's death, two of whom have been released.

Videos taken by witnesses on mobile phone cameras of the aftermath of the killing were broadcast around the world, but police continue to appeal for witnesses to the attack and for anyone who knew either of the two suspects to come forward.

An impromptu memorial to the soldier has been created close to where he was killed. People have paid their respects by leaving flags, football shirts and hundreds of bunches of flowers.

The Queen visited Woolwich barracks on Friday morning, during which time she was due to meet officers and soldiers associated with Rigby. The visit was a long-standing commitment but a Buckingham Palace spokeswoman said that "the Queen was privately acknowledging the events of last week".